When using both Gmail and Google Apps (with my own domain) for email, how can I get the Gmail account to send outgoing mail through email-account@gmail.com and have the user@example.com go through its own outbound server?

I want to have each of the accounts send and deliver from their respective accounts.

Currently they both are going out through the same server/domain, in this case user@example.com.

If you're using a desktop client, you'll need to make a second outgoing (smtp) account (not the best word, but it describes it). I'm not sure how you would do that (I haven't used a desktop client for a while, but the google help page is where I got my instructions for doing it from.

If you're using Gmail's internal multiple accounts system you just go to gear -> mail settings -> accounts and import and in the send mail as section, choose "Reply from the same address the message was sent to".

If you've got that setup, and your problem is that you just can't send it from the apps address through Gmail without using multiple accounts/signing in/out, in the compose window, there's a dropdown box for the FROM field with all of the accounts you've setup to send mail as.